In an trapezium of ABCD with diagonals that intersect in the middle, are all the triangles similar?

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writer2019

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Hey Guys, in this trapezium I know that AB is parallel to DC. I can also see that abe = edc because they are alternate angles. and dec = bea because they are vertically opposite.
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If this is the case, is it possible for the the adjacent triangles (DEC and CEB) to be similar as well?
In my honest opinion, I think they could because maybe the angles might be similar as it was as I mentioned above but I'm not overly confident. Would it be ade = bce ? Would my thinking of approaching this be correct
 
writer2019 said:
If this is the case, is it possible for the the adjacent triangles (DEC and CEB) to be similar as well?
In my honest opinion, I think they could because maybe the angles might be similar as it was as I mentioned above but I'm not overly confident. Would it be ade = bce ? Would my thinking of approaching this be correct
Click to expand...

See the red highlight, did you intend to write "ADE"?

If you meant "ADE" then visualize (or maybe sketch) vertex A having shifted to the left so that ADE becomes isosceles (obviously maintaining AB parallel to DC). Leave B,C,D in original positions. Does this help to answer your question?
 
Sorry I forgot to mention the fact that it is an irregular trapezium (which means that none of the sides are equal). , So does that mean since abe = edc does that mean that ade = ceb? So would that mean that the adjacent triangles (ADE and CEB) are similar?
 
Cubist said:
See the red highlight, did you intend to write "ADE"?

If you meant "ADE" then visualize (or maybe sketch) vertex A having shifted to the left so that ADE becomes isosceles (obviously maintaining AB parallel to DC). Leave B,C,D in original positions. Does this help to answer your question?
Click to expand...
Sorry I forgot to mention the fact that it is an irregular trapezium (which means that none of the sides are equal). , So does that mean since abe = edc does that mean that ade = ceb? So would that mean that the adjacent triangles (ADE and CEB) are similar?
 
Each of the four triangle has three angles. Two plane triangles are similar if the measures of two angles in one equal the measures of two angles in the other.

What do you conclude?
 
JeffM said:
Each of the four triangle has three angles. Two plane triangles are similar if the measures of two angles in one equal the measures of two angles in the other.

What do you conclude?
Click to expand...

I would conclude that they would be similar, but in the question I received its asks weather ADE and CEB would be similar without taking additional measurements. When I checked online, it said it wouldn't be similar due to the lack of conditions. I'm getting confused now by this, but I feel like they wouldn't be similar as seen with this site: http://jwilson.coe.uga.edu/EMT668/EMAT6680.2004.SU/Bird/emat6690/trapezoid/trapezoid.html
But the issue/my lack of understanding with what they have said is that I don't know if their trapeziums are irregular
 
writer2019 said:
I would conclude that they would be similar, but in the question I received its asks weather ADE and CEB would be similar without taking additional measurements. When I checked online, it said it wouldn't be similar due to the lack of conditions. I'm getting confused now by this, but I feel like they wouldn't be similar as seen with this site: http://jwilson.coe.uga.edu/EMT668/EMAT6680.2004.SU/Bird/emat6690/trapezoid/trapezoid.html
But the issue/my lack of understanding with what they have said is that I don't know if their trapeziums are irregular
Click to expand...

It would be very helpful if you would quote the exact wording of the problem you were given. It keeps changing.

There is a huge difference between "is it possible for the the adjacent triangles (DEC and CEB) to be similar as well?" and "whether ADE and CEB would be similar without taking additional measurements". There may be special cases in which one of those pairs of triangles could be similar; but certainly the given data are insufficient to conclude that they necessarily are; in fact, they are not similar in general.
 
Dr.Peterson said:
It would be very helpful if you would quote the exact wording of the problem you were given. It keeps changing.

There is a huge difference between "is it possible for the the adjacent triangles (DEC and CEB) to be similar as well?" and "whether ADE and CEB would be similar without taking additional measurements". There may be special cases in which one of those pairs of triangles could be similar; but certainly the given data are insufficient to conclude that they necessarily are; in fact, they are not similar in general.
Click to expand...
The exact wording of the question is: are the triangles adjacent to the proved also similar?
I was kind of putting it in context so people could understand it, I apologize for my English. But the proved is talking about triangle BEA and triangle DEC which was proved similar through AAA. But if my thinking is correct after some research: they are not similar as you said probably due to the fact the legs of the trapezium have different lengths thus changing the angle making them not similar
 
Cubist said:
visualize (or maybe sketch) vertex A having shifted to the left so that ADE becomes isosceles (obviously maintaining AB parallel to DC). Leave B,C,D in original positions. Does this help to answer your question?
Click to expand...

similar.png

Do they seem similar now?

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It might be a nice challenge to do this mathematically by assigning variables to some angles, and then calculating the remaining ones in terms of those variables. This would also tell you the special cases when ADE is similar to BCE (and/or CBE)
 
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